Asymptotic behavior of elementary solutions of transient generalized diffusion equations
Let \({\mathcal G}=(d/dm)(d/dx)\) be a generalized diffusion operator on an interval \(S=(\ell_ 1,\ell_ 2)\), -\(\infty \leq \ell_ 1<\ell_ 2\leq +\infty\), and p(t,x,y) the elementary solution of the generalized diffusion equation \[ \partial u(t,x)/\partial t={\mathcal G} u(t,x),\quad t>0,\quad x\in S. \] Let \(0<\rho <1\) and L be a slowly varying function. Assume one of the following conditions (A.i): (A.1) \(| \ell_ i| <\infty\), \(i=1,2\), there exists the limit \[ \theta \equiv \lim_{x\to \infty}| m(\ell_ 2-1/x)/m(\ell_ 1+1/x)| \in [0,\infty), \] and m satisfies \[ (1)\quad \lim_{x\to \infty}| m(\ell_ 1+1/x)| /x^{1/\rho +1}L(x)=1. \] (A.2) \(\ell_ 1>-\infty\), \(\ell_ 2=\infty\), there exists the limit \[ \lim_{x\to \infty}| x^ 2m(x)/m(\ell_ 1+1/x)| =0, \] and (1) holds. (A.3) \(\ell_ 1>-\infty\), \(\ell_ 2=\infty\), there exists the limit \[ \tau \equiv \lim_{x\to \infty}| x^ 2m(x)/m(\ell_ 1+1/x)| \in (0,\infty], \] and m satisfies \[ (2)\quad \lim_{x\to \infty}m(x)/x^{1/\rho -1}L(x)=1. \] Then it holds that \(\lim_{t\to \infty}t^{\rho +1}K(t)p(t,x,y)=c(x,y)\) uniformly in \(x,y\in [a,b]\), \(a,b\in S\), where c(x,y) is an explicit constant depending only on \(\rho\),x,y,\(\theta\),\(\tau\), and K is a slowly varying function such that \[ \lim_{x\to \infty}K(x)^{1/\rho}L(x^{\rho}K(x))=\lim_{x\to \infty}L(x)^{\rho}K(x^{1/\rho}L(x))=1. \] In particular, if \(\ell_ 1+\int^{0}_{\ell_ 1}(y-\ell_ 1)^{-1/2}m(y)dy>-\infty\), \(\ell_ 2=\infty\) and (2) is fulfilled, then \[ \lim_{t\to \infty}t^{\rho +1}K(t)p(t,x,y)=c(\rho)(x-\ell_ 1)(y-\ell_ 1) \] uniformly in \(x,y\in (\ell_ 1,a]\), \(a\in S\), where c(\(\rho)\) is a constant depending only on \(\rho\).
scientific article; zbMATH DE number 4082662